## Abstract

Let H be a stratum of translation surfaces with at least two singularities, let m_{H} denote the Masur-Veech measure on H, and let Z_{0} be a flow on (H, m_{H}) obtained by integrating a Rel vector field. We prove that Z_{0} is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces (L, m_{L}), where L ⊂ H is an orbit-closure for the action of G = SL_{2} (R) (i.e., an affine invariant subvariety) and m_{L} is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of Z_{0} with respect to any of the measures m_{L} is zero.

Original language | English |
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Article number | e7 |

Journal | Forum of Mathematics, Pi |

Volume | 12 |

DOIs | |

State | Published - 2 Apr 2024 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Statistics and Probability
- Mathematical Physics
- Geometry and Topology
- Discrete Mathematics and Combinatorics